Generalization of the noise model for time-distance helioseismology

D. Fournier, Laurent Gizon, T. Hohage, A. C. Birch

Research output: Contribution to journalArticle

Abstract

Context. In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here, we consider travel times and also products of travel times as observables. They contain information about the statistical properties of convection in the Sun. Aims. We derive analytic formulae for the noise covariance matrix of travel times and products of travel times. Methods. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process that is stationary and Gaussian. We generalize the existing noise model by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. Results. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to 1 /T, 1 /T2, and 1 /T 3, where T is the duration of the observations. For typical observation times of a few hours, the term proportional to 1 /T2 dominates and Cov [τ1τ2, τ3τ4] ≈ Cov [τ1, τ3] Cov [τ24] + Cov [τ14] Cov [τ23], where the τi are arbitrary travel times. This result is confirmed for p1 travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. Conclusions. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis.

Original languageEnglish (US)
Article numberA137
JournalAstronomy and Astrophysics
Volume567
DOIs
StatePublished - Jan 1 2014

Fingerprint

helioseismology
travel time
travel
wave field
matrix
products
moments
holography
random wave
solar interior
random processes
homogeneity
convection
diagram

Keywords

  • Convection
  • Methods: data analysis
  • Methods: statistical
  • Sun: granulation
  • Sun: helioseismology
  • Sun: oscillations

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science

Cite this

Generalization of the noise model for time-distance helioseismology. / Fournier, D.; Gizon, Laurent; Hohage, T.; Birch, A. C.

In: Astronomy and Astrophysics, Vol. 567, A137, 01.01.2014.

Research output: Contribution to journalArticle

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abstract = "Context. In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here, we consider travel times and also products of travel times as observables. They contain information about the statistical properties of convection in the Sun. Aims. We derive analytic formulae for the noise covariance matrix of travel times and products of travel times. Methods. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process that is stationary and Gaussian. We generalize the existing noise model by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. Results. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to 1 /T, 1 /T2, and 1 /T 3, where T is the duration of the observations. For typical observation times of a few hours, the term proportional to 1 /T2 dominates and Cov [τ1τ2, τ3τ4] ≈ Cov [τ1, τ3] Cov [τ2,τ4] + Cov [τ1,τ4] Cov [τ2,τ3], where the τi are arbitrary travel times. This result is confirmed for p1 travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. Conclusions. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis.",
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N2 - Context. In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here, we consider travel times and also products of travel times as observables. They contain information about the statistical properties of convection in the Sun. Aims. We derive analytic formulae for the noise covariance matrix of travel times and products of travel times. Methods. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process that is stationary and Gaussian. We generalize the existing noise model by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. Results. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to 1 /T, 1 /T2, and 1 /T 3, where T is the duration of the observations. For typical observation times of a few hours, the term proportional to 1 /T2 dominates and Cov [τ1τ2, τ3τ4] ≈ Cov [τ1, τ3] Cov [τ2,τ4] + Cov [τ1,τ4] Cov [τ2,τ3], where the τi are arbitrary travel times. This result is confirmed for p1 travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. Conclusions. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis.

AB - Context. In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here, we consider travel times and also products of travel times as observables. They contain information about the statistical properties of convection in the Sun. Aims. We derive analytic formulae for the noise covariance matrix of travel times and products of travel times. Methods. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process that is stationary and Gaussian. We generalize the existing noise model by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. Results. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to 1 /T, 1 /T2, and 1 /T 3, where T is the duration of the observations. For typical observation times of a few hours, the term proportional to 1 /T2 dominates and Cov [τ1τ2, τ3τ4] ≈ Cov [τ1, τ3] Cov [τ2,τ4] + Cov [τ1,τ4] Cov [τ2,τ3], where the τi are arbitrary travel times. This result is confirmed for p1 travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. Conclusions. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis.

KW - Convection

KW - Methods: data analysis

KW - Methods: statistical

KW - Sun: granulation

KW - Sun: helioseismology

KW - Sun: oscillations

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